Tacticity Effect on the Conformational Properties of Polypropylene and

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Tacticity Effect on the Conformational Properties of Polypropylene and Poly(ethylene−propylene) Copolymers Panagiotis-Nikolaos Tzounis, Dora V. Argyropoulou, Stefanos D. Anogiannakis, and Doros N. Theodorou*

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School of Chemical Engineering, National Technical University of Athens, GR 15780 Athens, Greece ABSTRACT: Tacticity is one of the most important features of stereoregular polymers, with significant impact on morphology and on a variety of properties such as conformational, thermal, rheological, mechanical, etc. In this work we focus on tacticity effects on the conformations and more specifically on the unperturbed dimensions of single polypropylene (PP) homopolymer and both block and random poly(ethylene−propylene) copolymer chains. The equilibration of all chains is achieved by applying the single-chain Monte Carlo algorithm of Tzounis et al. [Macromolecules 2017, 50, 4575−4587]. In agreement with past studies, we find that tacticity has a significant effect on the stiffness of PP homopolymer. The characteristic ratio exhibits a nonmonotonic dependence on the fraction of meso dyads along the PP chains, which results from two competing mechanisms, revealed by analysis of the torsional states of the PP chain backbones. A simple theoretical model is developed to describe the dependence of the stiffness of poly(ethylene-block-propylene) copolymer on its propylene content and on the tacticity of its PP block. Finally, the effect of both tacticity and propylene content on the stiffness of poly(ethylene-randompropylene) chains, used to model ethylene−propylene monomer (EPM) materials, is found to be in very good qualitative agreement with available rotational isomeric state (RIS) model predictions.

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etc., out of copolymers of the same composition.44−46 Therefore, tacticity of the blocks is a very important design parameter in block copolymer applications, such as directed self-assembly (DSA) lithography.44−48 Tacticity affects the distribution of torsion angles along the backbones of polymer chains and as a result has a significant influence on chain stiffness. The effect of tacticity on chain stiffness, which is usually quantified through Flory’s characteristic ratio, C∞, or through the Kuhn length, b, has been a subject of many theoretical and experimental studies. Several theoretical works have extracted the dependence of stiffness on tacticity for various polymers such as polystyrene, poly(methyl methacrylate), and poly(lactide)49−53 through rotational isomeric state (RIS) model calculations. Experimentally, the effect of tacticity on the unperturbed dimensions of various polymer chains has been studied through small-angle neutron scattering (SANS) measurements in the melt54,55 and intrinsic viscosity measurements in Θ-solutions.56,57 The dependence of chain stiffness on tacticity is important because chain stiffness is related to a plethora of polymer properties, such as viscoelastic properties,24,58,59 and is also a key input parameter to many coarse-grained models60−64 and self-consistent meanfield calculations.65−69

INTRODUCTION Tacticity or chirality of polymer chains, which is defined as the arrangement of adjacent chiral centers along a polymer chain, is one of the most important parameters in the design of macromolecules, as it strongly affects the properties of the materials obtained from them. Tacticity is usually identified by the sequence of the so-called dyads, which consist of two sequential chiral units along the polymer chain. There are two kinds of dyads: meso (m) dyads, consisting of two chiral units of the same stereochemical configuration, and racemo (r) dyads, where the two chiral units have opposite configurations. Experimentally, the tacticity of macromolecules is usually characterized by 13C NMR spectroscopic techniques,1−6 while in simulations it is usually identified by geometric criteria.7−15 Tacticity plays a significant role in many properties of polymers. Dynamical properties such as the self-diffusion coefficient Dcm16 and the glass transition temperature Tg,17−21 rheological properties such as the entanglement molar mass Me and the plateau modulus Ge,22−27 and the crystallization28−30 as well as the permeability31 of chiral homopolymers are strongly affected by their tacticity. They have been extensively studied in the literature by both experiments and simulations. Furthermore, the miscibility of various polymer blends17,32−43 is determined by the tacticity of the mixed chains. The effect of tacticity on polymer chain miscibility is also reflected in the different microphases in which block copolymers usually selforganize. Different tacticities can lead to the formation of different microphases such as lamellae, cylindrical, hexagonal, © XXXX American Chemical Society

Received: May 25, 2018 Revised: August 5, 2018

A

DOI: 10.1021/acs.macromol.8b01099 Macromolecules XXXX, XXX, XXX−XXX

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(b) the theoretical model we have developed to link the stiffness of unperturbed poly(ethylene-block-propylene) chains with their chemical composition and tacticity, and (c) the dependence of stiffness of poly(ethylene-random-propylene) chains on the tacticity of their propylene sequences, based on a statistical analysis of these sequences.

In this work we choose to systematically study the effects of tacticity on the conformational properties of both homopolymer and copolymer unperturbed chains using single-chain Monte Carlo simulations. Our main target is to identify the microscopic mechanisms through which the chirality of polymer chains affects their stiffness. A very promising candidate in this study was polypropylene (PP) homopolymer due to its simple chemical constitution and the plethora of different conformations it adopts (i.e., 31 helical conformation with s(3/1)1 symmetry for isotactic, 21 helical conformation with s(2/1)2 symmetry for syndiotactic, and random coil conformation for atactic).27,44,70−72 The stiffness and the conformations of single unperturbed PP chains and their dependence on tacticity at various temperatures have been studied in many works through RIS calculations,73−77 and the results have been compared against the corresponding experimental estimates through SANS58,78 and/or intrinsic viscosity measurements.79 Unfortunately, none of these works provide a microscopic explanation of how tacticity affects stiffness and what are the mechanisms in action. A more comprehensive study of the effect of tacticity on the structure and the dynamics of PP melts was made by Antoniadis et al.80,81 Performing molecular dynamics (MD) simulations on isotactic, syndiotactic, and atactic PP melts, they identified the tacticity effects on the glass transition temperature and the segmental dynamics of PP chains. They also performed a systematic analysis of the transitions occurring between different isomeric states as a function of PP chain tacticity, which are mainly responsible for tacticity effects on segmental dynamics. Apart from PP homopolymers, we also choose to study how tacticity affects the conformational properties of poly(ethyleneblock-propylene) copolymers and poly(ethylene-random-propylene) copolymers, the latter constituting a model for ethylene−propylene monomer (EPM) materials. EPMs are widely used mainly as rubbers in applications such as compatibilizers,82 tires,83 electrical and thermal insulations,83,84 and pipes.83,85 Mark86 studied how the unperturbed EPM chain dimensions are affected by both the tacticity and the propylene content using RIS calculations. However, a physical explanation of how propylene content or tacticity affects the stiffness of these random copolymers is still missing. Experimentally, the stiffness of EPMs at various propylene contents is estimated from intrinsic viscosity measurements in Θ-solutions87 and SANS measurements23,58,88 in polymer melts, but unfortunately, we were unable to find any experimental work which systematically studies how the tacticity of the propylene units would affect the stiffness of EPM chains. However, we found in the literature that the propylene sequences along EPM chains are short; i.e., their average length is approximately 2−2.5 propylene units,23,82 suggesting that tacticity should not play a significant role in their conformations. In any case, in the Results section, we will also study the impact of the length of propylene unit sequences on EPM chain conformations. The paper is organized as follows. First we describe in detail the systems we studied and the force field we applied in our simulations. Then, we briefly review the single unperturbed MC simulation methodology we applied to produce long equilibrated MC trajectories of single unperturbed chains. Finally, we present and discuss the results of our study and more specifically (a) the two mechanisms we have identified as affecting the tacticity-dependent stiffness of pure PP chains,

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SYSTEMS STUDIED In this work, we have studied single unperturbed chains of (i) polyethylene (PE), (ii) polypropylene (PP), (iii) poly(ethylene-block-propylene) copolymers, henceforth termed PE-b-PP, and (iv) random copolymers of ethylene and propylene, termed EPM in the following. The molecular characteristics of these systems are described in detail below. Polyethylene (PE). Using the Amorphous Builder of the MAPS suite, developed by Scienomics,89 we generated an initial configuration for a single linear PE chain whose chemical constitution is CH3−CH2−(CH2−CH2)499−CH3. In all PE calculations we have applied the Transferrable Potential for Phase Equilibria United Atom (TraPPE-UA) force field,90 wherein hydrogen atoms are fused with the carbon atoms to which they are connected. Therefore, the generated PE chain consists of 1001 united atoms (999 CH2 and 2 CH3 united atoms). The version of the TraPPE-UA we applied was the one used in the work of Curro and Grest,91,92 with a harmonic bond stretching potential instead of fixed bond lengths. Polypropylene (PP). The next systems we studied were a series of PP chains with various probabilities (fractions) of meso dyads, pm. In particular, beginning from a single isotactic PP (i-PP) chain (pm = 1) generated by the Amorphous Builder of the MAPS suite,89 we rotated some of its methyl branches to produce ten new PP chains with pm = 0.9, 0.8, ..., 0.0, where pm = 0.0 and pm = 0.5 correspond to the syndiotactic PP (s-PP) and atactic PP (a-PP) chains, respectively. All meso and racemo (tacticity) sequences were Bernoullian. All PP chains were constructed with head-to-tail configurations, where no propylene inversion was allowed. The united-atom force field we applied was that of Curro and Grest,91,92 already adopted for PE. All PP chains consisted of 1501 united atoms in total. The backbone of each chain comprised 1001 united atoms (500 CH, 499 CH2, and 2 terminal CH3), while there were 500 methyl substituents. In other words, the chemical constitution of the chains was CH3−[CH(CH3)−CH2]499−CH(CH3)2. Poly(ethylene−propylene) Block Copolymers (PE-bPP). Apart from PE and PP homopolymer chains, we have also generated and studied three series of PE-b-PP copolymer chains. Each series consisted of nine PE-b-PP chains with mole fraction xP = 0.1, 0.2, ..., 0.9 propylene units. In the first series all dyads in the PP block are meso (i-PP block), in the second all are racemo (s-PP block), and in the last series there are 50% meso and 50% racemo dyads (a-PP block) with Bernoullian tacticity sequences. All PE-b-PP chains consist of the same number of backbone bonds n = 1000 as the pure PP chains described above. In other words, the chemical constitution of the block copolymer chains was CH3−(CH2−CH2)500(1−xp)− [CH(CH3)−CH2]500xp−1−CH(CH3)2, for 0 < xp < 1. We applied the same force field as used for the pure PE and PP chains.91,92 A synopsis of the number of united atoms and the number of ethylene and propylene units of the PE-b-PP chains is shown in Table 1. Ethylene−Propylene Random Copolymers (EPMs) and Alternating Ethylene−Propylene Copolymer (altB


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successfully applied to a variety of polymer systems with different chemical constitution. The basic idea of the algorithm is that beginning from the initial configuration of a single polymer chain and performing simple Monte Carlo moves, which are being accepted or rejected according to a Metropolis acceptance criterion, the chain can be fully equilibrated and adopt its unperturbed dimensions. In this work, only the single atom displacement, f lip atom, and rotate strand (pivot) moves for the backbone atoms and the f lip branch move for the branches are performed. Because of the fact that only single unperturbed chains and not entangled polymer melt bulk phases are simulated here, there is no need for concerted rotation, configurational bias, or other computationally demanding moves to be applied. On the other hand, the rotate strand (pivot) move, which is not applicable to Monte Carlo simulations of condensed phases, is one of the most effective moves here and is attempted with a high frequency. In particular, for the PP homopolymer and poly(ethylene−propylene) copolymer chains the attempt frequencies are 10% for the single atom displacement, 10% for the f lip atom, 70% for the rotate strand (pivot) move applied to the backbone atoms, and 10% for the f lip branch moves. Especially for the PE chains the attempt frequencies are 10% for the single atom displacement move, 10% for the flip atom move, and 80% for the rotate strand (pivot) move. In both cases we choose to have high percentages for the rotate strand (pivot) move due to the fact that it is the most effective move in decorrelating our single-chain conformations. The most important feature of the algorithm lies in its treatment of intermolecular nonbonded interactions, which ensures that the sampled polymer chains are indeed unperturbed and representative of those encountered in a melt. According to Flory’s definition,94 in a single unperturbed chain there are only “local” interactions. The algorithm provides a methodology and a simple criterion to determine the optimum range of intermolecular interactions within which all interactions can be considered as “local”. This range is estimated by systematically varying the maximal topological

Table 1. Synopsis of the Chemical Constitution of PE-b-PP, EPM, and alt-PEP Systems xP

no. of atoms

no. of ethylene units

no. of propylene units

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1051 1101 1151 1201 1251 1301 1351 1401 1451

450 400 350 300 250 200 150 100 50

50 100 150 200 250 300 350 400 450

PEP). Finally, we studied five series of EPM chains. Each series consists of nine EPM chains with xP = 0.1, 0.2, ..., 0.9 propylene units and corresponds to one of the following probabilities of meso dyads pm = 0, 0.3, 0.5, 0.7, and 1. In addition, a single alternating poly(ethylene-alt-propylene) chain (alt-PEP) was also generated for comparison purposes. Both EPM and alt-PEP chains have the same number of backbone bonds, n = 1000, as the PP and PE-b-PP chains we described previously. The TraPPE-UA force field was once again applied for all chains.91,92 A synopsis of the number of united atoms as well as the number of ethylene and propylene units of both EPM chains and alt-PEP chain is shown in Table 1.

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METHODOLOGY As we have already mentioned in the Introduction, the main goal of this work is to study the effect of tacticity on the conformational properties (Flory’s characteristic ratio, Kuhn length, etc.) of single unperturbed chains of chiral homopolymers as well as block and random copolymers with various constitutions. The ensembles of single unperturbed chains whose conformational properties were calculated were created and equilibrated by applying the Metropolis Monte Carlo algorithm developed by Tzounis et al.,93 which has been

Figure 1. (a) Flory’s characteristic ratio, C∞, plotted as a function of Δnpair for the syndiotactic (pm = 0.0), atactic (pm = 0.5), and isotactic (pm = 1.0) PP chains. (b) Schematic representation of four PP chains. With red ellipses we mark the monomers whose interactions with the central monomer, indicated with the purple dashed rectangle, are considered as local and therefore taken into account in the MC sampling, for each one of the cases Δnpair = 1, 2, 3, and 4. C


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Macromolecules distance of repeat units (Δnpair) between which nonbonded forces are active, until a maximum in stiffness (expressed by Flory’s characteristic ratio) is achieved. As described in the Systems Studied section, the initial configurations of the single chains, which are the input of the MC algorithm, were basically generated using the amorphous builder plugin of MAPS suite.89 However, the desired probabilities of meso dyads (pm), in both homopolymer and copolymer chains, were achieved by applying a home-built algorithm which alters the generated conformations by rotating the methyl branches, implementing a Bernoullian distribution of dyads, until the desired pm is reached.

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RESULTS AND DISCUSSION In this section we present and discuss the results of our study of the effect of tacticity on the unperturbed dimensions of PP homopolymer chains as well as of PE-b-PP and EPM copolymer chains with various PP contents. Note that in all figures error bars are very small, comparable to the width of the lines, and therefore are omitted. PP. The first set of systems we studied was a series of PP chains characterized by Bernoullian distribution of dyad types with probability of meso dyads pm = 0, 0.1, 0.2, ..., 1.0. For each system we perform single unperturbed chain MC simulations at constant temperature T = 450 K, applying the algorithm we described in the Methodology section, for approximately 1 billion MC steps and for different Δnpair values to determine the one where stiffness, expressed by Flory’s characteristic ratio, is maximized. In Figure 1a, we plot the characteristic ratio C∞ as a function of Δnpair for the three homopolymer syndiotactic (pm = 0.0), atactic (pm = 0.5), and isotactic (pm = 1.0) PP chains. We note that C∞ is maximized at Δnpair = 2 for the s-PP and a-PP chains and at Δnpair = 3 for the i-PP chain. In Table 2 we show the values of Δnpair which maximize C∞ for each pm.

Figure 2. Flory’s characteristic ratio, Cn, plotted as a function of the number of backbone bonds, n, for PE, s-PP, a-PP, and i-PP at 450 K.

stiffer than both i-PP (C∞ = 6.2) and a-PP (C∞ = 6.0), while the last two are close to each other with i-PP being a little stiffer than a-PP. However, quantitatively, all C∞ values are overestimated by our simulation in comparison with the experimental ones; this is especially true of the C∞ value of sPP. The overestimation is in all likelihood due to the TraPPEUA force field we apply and, more specifically, to its torsional potential, which strongly affects the stiffness of polymer chains. In the previous work of Tzounis et al.93 the stiffness of i-PP using the same MC algorithm and at the same temperature, but this time applying the hybrid force field proposed by Logotheti and Theodorou,97−99 was estimated as C∞ = 5.9,93 smaller than the one we calculate here and closer to the experimental value. For comparison, we have also calculated the stiffness of a single unperturbed PE chain using the TraPPE-UA force field. We note that the red curve in Figure 2, which corresponds to PE chain, lies between the curves for s-PP and i-PP. From the plateau value we can estimate C∞ = 8.9 for PE, higher than the estimate of C∞ = 8.2 reported by Tzounis et al.,93 where the anisotropic united-atom (AUA) torsional potential proposed by Toxvaerd100 was applied, using the same MC algorithm. This overestimation of characteristic ratios for PE by the TraPPE-UA force field is well-known in the literature92,101,102 and has recently been discussed in detail in the work of Ramos et al.,103 where a comprehensive review of the estimation of various physical and thermodynamic quantities of PE through computer simulations is presented. In Figure 3, we plot the modified persistence length:

Table 2. Δnpair Values That Maximize C∞ for Each pm for the PE and PP Homopolymer Chains system

pm

Δnpair

system

pm

Δnpair

system

pm

Δnpair

PE PP PP PP

N/A 0.0 0.1 0.2

2 2 2 2

PP PP PP PP

0.3 0.4 0.5 0.6

2 2 2 2

PP PP PP PP

0.7 0.8 0.9 1.0

3 3 3 3 ⟨R2⟩

In Figure 2, we plot Flory’s characteristic ratio, Cn = 2 , nl where ⟨R2⟩ is the mean-square end-to-end distance of a single unperturbed chain, n is the number of backbone bonds, and l = 1.54 Å is the average backbone bond length, as a function of n, for a pure PE chain and three PP chains with three different tacticities: an isotactic PP (pm = 1), a syndiotactic PP (pm = 0), and an atactic PP (pm = 0.5). The curves of PE, s-PP, and a-PP, shown with red, black, and green lines, correspond to a value of Δnpair = 2, while the one of i-PP, shown with a blue line, corresponds to Δnpair = 3. In the limit of long chains (n → ∞), ⟨R2⟩ ∼ n, and Cn reaches a plateau value, C∞, for each polymer. We note that the s-PP chain is the stiffest of all with C∞ = 11.71, while the stiffnesses of both i-PP (C∞ = 7.12) and a-PP chains (C∞ = 6.59) are much smaller and close to each other. Qualitatively, these estimates of the stiffness of PP chains are in agreement with experimental results from SANS measurements,58,95,96 according to which s-PP (C∞ = 9.12) is much

⟨l(pk)⟩ ≡

rk ·R |rk|

, as defined in the works of Connolly et

104,105

and Yethiraj et al.,106 where R is the end-to-end vector al. of the chain and rk is the bond vector of the k-th backbone bond versus k for the s-PP and i-PP chains. Note that k = 1 refers to the first backbone bond of the chain, and therefore ⟨l(1) p ⟩ corresponds to the original Flory definition of persistence length. Because of the fact that in this type of plot for each k value only a single value of l(k) is obtained from a given p unperturbed chain configuration, a very large sample of single unperturbed PP chains is needed to reduce the noise of the plot. For this reason we limit this analysis only to the s-PP (pm = 0) and i-PP (pm = 1) chains. In Figure 3, chain end effects on the conformation and the tacticity of PP chains are clearly reflected. We note that ⟨l(k) p ⟩ displays a strong dependence on k near chain ends, but at approximately k = 50 and k = 950 (i.e., D


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for pm = 0.038527 for s-PP, 0.502107 for a-PP, and 0.990527 for i-PP, denoted by red circles in Figure 4, we choose to compare our estimations with the results obtained from theoretical rotational isomeric state (RIS) model calculations,74,108,109 shown in Figure 4 as colored lines. The same nonmonotonic behavior is also evident in the works of Allegra et al.108 and Alfonso et al.,109 who found a minimum C∞ value at approximately pm = 0.45 and 0.57, respectively. On the other hand, Suter and Flory74 predicted a monotonic decrease of C∞ as pm increases. Furthermore, for pm ≤ 0.5 our estimates for C∞ are in good agreement with the values calculated in the work of Suter and Flory,74 while for pm > 0.5 they progressively deviate from that work and are in good agreement with the ones calculated by Alfonso et al.108 Unfortunately, none of the above-mentioned theoretical works provide a physical explanation regarding the appearance of a minimum stiffness. In this direction, we performed an analysis of the single unperturbed PP chain conformations we simulated, based on the torsional state sequences of their backbones. First of all, we extract the normalized distributions of torsion angles along the backbone of the PP chains, shown in Figure 6a, and we identify the three torsional states, trans (t), gauche (g), and gauche bar (g̅). In particular, the gauche bar (g̅) state corresponds to the torsional angle range 0° ≤ φ < 125°, the trans (t) state to 125° ≤ φ < 245°, and finally the gauche (g) state to 245° ≤ φ ≤ 360°. The bounds of the ranges we use are determined from the minima of the normalized distributions of Figure 6a, i.e., φ = 0°, 125°, 245°, and 360° correspond to the first, second, third, and fourth minima of the distributions. The definitions of these torsional states using Newman projections are shown in Figure 5b. Here, we should mention that in the RIS model for PP developed by Suter and Flory74 five torsional states are defined, i.e., trans (t), trans star (t*), gauche (g), gauche star (g*), and gauche bar (g̅). In our continuous simulations, trans star (t*) and gauche star (g*) are merged with trans (t) and gauche (g), respectively, giving rise to the corresponding asymmetric trans (t) and gauche (g) peaks. The same behavior is evident in other works with continuous simulations, either MC or MD, of PP melt chains.80,81,97 We notice that as pm decreases (moving from i-PP to s-PP) the probability of the trans state increases, while the probability of the gauche (g) state decreases, both indicating that PP chains become stiffer, with higher characteristic ratios. On the other hand, gauche bar (g̅) probabilities are very small, almost negligible, due to the fact that gauche bar states are energetically unfavorable. Figure 6a provides no evidence of a nonmonotonic behavior such as the one we show in Figure 4. In view of this, we continue our analysis by calculating, in Figure 6b, the average percentages of trans, pt, and gauche, pg, torsional angles as a function of pm. We note that as pm increases (moving from s-PP to i-PP) pt decreases and pg increases, as expected from the distributions of Figure 6a. Around pm = 0.7 (marked by a dashed vertical line), which corresponds to the probability where the minimum stiffness appears in Figure 4, we simulated and analyzed PP chains with pm = 0.62, 0.64, ..., 0.9 to easily track any change in the behavior of pt and pg,. Unfortunately, there is no noticeable change in the slope of either curve. Similar behavior is observed in Figure 6c, where the fractions of tt, gg, and tg pairs of torsion angles, ptt, pgg, and ptg, respectively, are plotted against pm (gt pairs are lumped together with tg pairs). In the inset of Figure 6c, where a zoomed version of the pgg versus pm

Figure 3. Modified persistence length ⟨l(k) p ⟩ plotted as a function of the index number of the backbone bond vector k for the s-PP and i-PP chains.

50 backbone bonds far from chain ends) it reaches an asymptotic value. This behavior is in agreement with the one shown in Figure 2, where Flory’s characteristic ratio, Cn, reaches its plateau value at approximately n = 100 backbone bonds. To investigate systematically how tacticity affects the stiffness of PP chains, we plot C∞ as a function of the probability of meso dyads, pm, in Figure 4 (black squares). A

Figure 4. Characteristic ratio, C∞, as a function of the probability of meso dyads, pm, for the PP homopolymer chains. Black squares correspond to our simulation results, and red circles correspond to the experimental values in the work of Fetters et al.58,95,96 The orange, green, and blue curves correspond to the RIS calculations of Suter and Flory,74 Alfonso et al.,109 and Allegra et al.,108 respectively. In the inset the statistical (Kuhn) segment length, b, is plotted as a function of the probability of meso dyads, pm. Black squares correspond to our simulation results, while red circles correspond to the experimental measurements of Fetters et al.58,95,96

schematic representation of a meso (m) and a racemo (r) PP dyad, m, is shown in Figure 5a. We note that the dependence of C∞ on pm is not monotonic and a minimum stiffness appears at approximately pm = 0.7. As expected, the same behavior is also present in the inset of Figure 4, where the Kuhn length b=

⟨R2⟩ nl cos(θ / 2)

=

C∞l , cos(θ / 2)

with π−θ being the average angle

between two sequential backbone bonds, is plotted versus pm. At approximately pm = 0.7 there is a minimum of b which corresponds to the minimum stiffness. Because of the lack of experimental measurements of C∞ for various pm, except those E


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Figure 5. (a) Schematic representation of a meso, m, and a racemo, r, PP dyad. (b) Newman projections of three states g̅, t, and g. (c) Snapshot of helical conformation of i-PP chain, which corresponds to the tgtgtg... torsional state sequence.

Figure 6. (a) Normalized backbone torsional angle distributions for PP chains with different probabilities of meso dyads, pm = 0, 0.1, ..., 1. Trans (t), gauche (g), and gauche bar (g̅) states have been identified. A zoom into the g̅ peak is shown in the inset of the plot. (b) Average fractions of trans, pt, and gauche, pg, states versus pm denoted with black and red lines and symbols, respectively. (c) Fractions of tt (ptt), gg (pgg), and tg (ptg) pairs of torsion angles versus pm. In the inset a zoomed pgg versus pm is shown. (d) Average maximum length, expressed in number of torsion angles, of allmax max max trans ⟨nmax t ⟩, all-gauche ⟨ng ⟩, and all-trans-gauche ⟨ntg ⟩ sequences along the backbone of PP chains versus pm. In the inset a zoomed ⟨ng ⟩ versus pm plot is shown.

change in the slope at pm = 0.7, similar to the one shown in the inset of Figure 6c. However, the changes in the slopes of the corresponding plots we observed so far are only indications that a change in the conformation of PP chains occurs around pm = 0.7 but do not provide a concrete physical interpretation. A very informative quantity that will aid us in the direction of providing a physical meaning and uncovering the origins of minimum stiffness appearing in Figure 4 is ⟨nmax tg ⟩ which, as mentioned above, is plotted with blue line and symbols in Figure 6d and refers to the maximum length of tgtgtg... sequences along the backbone of PP chains. These sequences correspond to a helical conformation of the PP chain, as shown

plot is shown, one can notice a small change in the slope at pm = 0.7. Finally, in Figure 6d we plot the average maximum lengths, expressed in number of torsion angles, of all-trans (ttttt...), all-gauche (ggggg...), and all-trans-gauche (tgtgtg...) max max sequences ⟨nmax t ⟩, ⟨ng ⟩, and ⟨ntg ⟩, respectively, versus pm. As expected, as we move from s-PP to i-PP (with increasing pm), ⟨nmax t ⟩ decreases; i.e., all-trans sequences become shorter on average, leading to less stiff PP conformations. In addition, as pm increases, ⟨nmax g ⟩ decreases, i.e., all-gauche sequences become shorter; in general, ggggg... sequences are not favorable, max as reflected by the low values of ⟨nmax g ⟩. In ⟨nt ⟩ there a max monotonic behavior, while in ⟨ng ⟩ there is a noticeable F


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Macromolecules Table 3. Probabilities of Triads of Torsional States pm sequence

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ttt ttg, gtt tgt tgg, ggt gtg ggg

0.44 0.25 0.03 0.21 0.01 0.01

0.40 0.28 0.07 0.18 0.03 0.01

0.35 0.29 0.11 0.16 0.04 0.01

0.30 0.30 0.15 0.14 0.18 0.01

0.26 0.31 0.18 0.12 0.09 0.01

0.22 0.31 0.22 0.10 0.11 0.01

0.17 0.31 0.25 0.08 0.14 0.01

0.13 0.29 0.28 0.08 0.18 0.01

0.12 0.28 0.29 0.07 0.19 0.00

0.06 0.23 0.34 0.06 0.26 0.01

0.03 0.19 0.37 0.06 0.30 0.00

Table 4. Probabilities of Tetrads of Torsional States pm sequence

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

tttt tttg, gttt ttgt, tgtt tgtg, gtgt ttgg, ggtt tggt gttg tggg, gggt gtgg, ggtg gggg

0.35 0.17 0.06 0.01 0.19 0.10 0.04 0.01 0.01 0.00

0.30 0.19 0.11 0.03 0.16 0.09 0.04 0.01 0.02 0.00

0.25 0.19 0.16 0.06 0.13 0.07 0.05 0.01 0.03 0.00

0.20 0.19 0.19 0.10 0.10 0.06 0.05 0.01 0.03 0.00

0.16 0.19 0.23 0.13 0.08 0.05 0.06 0.01 0.04 0.00

0.13 0.17 0.24 0.18 0.06 0.04 0.07 0.01 0.04 0.00

0.09 0.16 0.26 0.24 0.05 0.04 0.07 0.01 0.04 0.00

0.06 0.13 0.25 0.31 0.04 0.03 0.08 0.01 0.04 0.00

0.05 0.12 0.25 0.33 0.03 0.03 0.08 0.01 0.04 0.00

0.02 0.08 0.21 0.47 0.02 0.03 0.08 0.01 0.04 0.00

0.01 0.04 0.17 0.55 0.02 0.03 0.07 0.01 0.04 0.00

Table 5. Probabilities of Pentads of Torsional States pm sequence

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ttttt ttttg, gtttt tttgt, tgttt tttgg, ggttt ttgtt ttgtg, gtgtt ttggt, tggtt ttggg, gggtt tgttg, gttgt tgtgt tgtgg, ggtgt tggtg, gtggt tgggt tgggg, ggggt gtttg gttgg, ggttg gtgtg gtggg, gggtg ggtgg ggggg

0.27 0.16 0.04 0.13 0.02 0.01 0.18 0.01 0.01 0.00 0.01 0.01 0.00 0.01 0.01 0.06 0.00 0.00 0.00 0.00

0.22 0.15 0.08 0.10 0.04 0.03 0.15 0.01 0.03 0.01 0.02 0.02 0.00 0.01 0.02 0.06 0.00 0.00 0.00 0.00

0.17 0.14 0.11 0.08 0.06 0.05 0.12 0.01 0.05 0.02 0.03 0.03 0.00 0.00 0.02 0.05 0.01 0.00 0.00 0.00

0.13 0.13 0.13 0.06 0.06 0.06 0.10 0.01 0.07 0.03 0.03 0.03 0.00 0.00 0.03 0.04 0.01 0.00 0.00 0.00

0.10 0.12 0.14 0.04 0.07 0.09 0.07 0.01 0.09 0.05 0.03 0.03 0.00 0.00 0.03 0.03 0.02 0.00 0.00 0.00

0.07 0.11 0.14 0.03 0.07 0.10 0.05 0.01 0.10 0.07 0.03 0.03 0.00 0.00 0.03 0.03 0.04 0.00 0.00 0.00

0.05 0.09 0.13 0.02 0.06 0.12 0.04 0.01 0.12 0.10 0.03 0.03 0.00 0.00 0.03 0.02 0.06 0.00 0.00 0.00

0.03 0.06 0.11 0.02 0.05 0.14 0.03 0.00 0.13 0.14 0.03 0.03 0.00 0.00 0.03 0.02 0.08 0.00 0.00 0.00

0.03 0.06 0.11 0.10 0.05 0.14 0.03 0.00 0.13 0.15 0.03 0.03 0.00 0.00 0.03 0.02 0.09 0.00 0.00 0.00

0.01 0.03 0.07 0.01 0.03 0.14 0.02 0.00 0.14 0.21 0.04 0.03 0.00 0.00 0.02 0.01 0.16 0.00 0.00 0.00

0.00 0.01 0.04 0.00 0.02 0.13 0.01 0.00 0.13 0.25 0.04 0.04 0.00 0.00 0.02 0.01 0.21 0.00 0.00 0.00

average length of all-trans sequences, decrease, leading to less stiff conformations. However, at the same time, more and more lengthy helical conformations (tgtgtg...) appear, tending to increase the stiffness of the chains. The first process dominates at pm ≤ 0.7, causing C∞ to decrease as pm increases (see Figure 4). The second process dominates for pm > 0.7, causing C∞ to increase as pm increases. Clearly, the value of pm where the second process begins to affect the stiffness of PP chains, if at all, depends on the force field we invoke to describe the interactions of PP atoms. In the RIS models of Alfonso et al.108 and Allegra et al.107 the second process begins to act at pm ≃ 0.5, lower than our estimate. On the other hand, in the RIS

in Figure 5c. We note that the average length of such helical sequences increases progressively with pm, but at pm = 0.7 there is an abrupt change in the slope of ⟨nmax tg ⟩ with pm, indicating that there are long enough segments with helical conformations present along the PP chain backbones to bring about an increase in the overall stiffness of the chains. Combining the latter microscopic picture with the fact that all-trans sequences become shorter (and also both pt and ptt decrease) as pm increases, we can conclude that there are two opposing processes acting simultaneously and determining how tacticity affects PP stiffness. In the first process, as one moves from s-PP to i-PP, the population of trans-states and, therefore the G


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Macromolecules model of Suter and Flory74 the second process is unable to cause a noticeable change in stiffness, and the first process dominates for all pm values. We have extended our analysis to longer sequences. In Tables 3, 4, and 5 we show the probabilities of all possible triads, tetrads, and pentads formed from the dominant t and g torsional states for various pm values. In each column of the tables the probabilities do not add up to 1 because sequences involving the improbable g̅ torsional state have not been included. In all tables we mark in bold the sequences of torsional states which correspond to the two highest probabilities. For low pm values (pm < 0.3), i.e., as we approach s-PP, the most favorable triad is ttt, consisting of sequential trans states. On the other hand, for higher pm values (pm > 0.7), i.e., as we approach i-PP, the most favorable triad is tgt, a part of the helical conformation tgtgtgtg.... Similarly, the most favorable tetrads and pentads of torsional states for pm < 0.3 are tttt and ttttt, while for pm > 0.7 they are tgtg and tgtgt, respectively. The analysis of higher order sequences of torsional states, such as triads, tetrads, and pentads, provides further evidence for the two opposing mechanisms which act together and were described previously. One could ask the question, which of the sequence lengths we have analyzed is the most impactful, i.e., better reflects the conformational preferences induced by the stereo sequence, while at the same time not being too long. The answer, in our opinion, is the tetrads. The contrast between the probabilities of the two most preferable tetrad sequences and the rest is higher than the corresponding contrast for triads or pentads. For example, in iPP (pm = 1.0) the probability of the tgtg tetrad, ptgtg = 0.55, is very high compared to the probabilities of all other sequences. Similar behavior is observed for other tacticities, pm. This contrast is less strong in pentads probably due to the higher number of sequences that can be adopted. Finally, in Figure 7 we calculate the temperature coefficient,

calculations of various RIS models, displayed with colored lines. RIS models also predict a minimum value of |κ| at pm = 0.30−0.50, which is in good agreement with our estimations. However, our |κ| values are underestimated compared to RIS predictions,73,74,86,108,109 possibly due to the TraPPE-UA force field we apply. On the other hand, experimental measurements using SANS95,110 do not betray any significant temperature dependence of C∞ (κ = −0.1 for a-PP and κ = 0 for i-PP); unfortunately, we were unable to find any experimental results for other PP tacticities. PE-b-PP. In this section we study the effect of tacticity on the stiffness of PE-b-PP copolymer chains with various propylene contents. A 3D representation of a PE-b-PP chain consisting of 500 ethylene and 500 propylene units (xP = 0.5) is shown in Figure 8, where the PE and PP blocks are denoted

Figure 8. Schematic representation of end-to-end vectors on an unperturbed chain of PE-b-PP consisting of 500 ethylene and 500 propylene units denoted with cyan and blue colors, respectively. The cyan vector corresponds to the end-to-end vector of PE block, RPE, the blue vector to the end-to-end vector of PP block, RPP, and finally the black vector to the end-to-end vector RPE‑b‑PP of the whole PE-bPP chain.

d ln(C )

κ = dT ∞ as a function of pm, denoted with black squares. κ is predicted to be negative. The behavior with pm is once again nonmonotonic, with a minimum in |κ| appearing at approximately pm = 0.40. We compare our results with the

with cyan and blue colors, respectively. As we described in the Systems Studied section, we generate three series of PE-b-PP chains with pm = 0, 0.5, and 1.0. Each series consists of nine PE-b-PP chains with xP = 0.1, 0.2, ..., 0.9. In the end we have 27 PE-b-PP chains with various xP and pm values. We choose all PE-b-PP chains to have the same backbone length as the PE and PP homopolymer chains we simulated and studied in the previous section, i.e. n = 1000 backbone bonds. For each system we perform single unperturbed chain MC simulations, at constant temperature T = 450 K, applying the algorithm we described in the Methodology section, for approximately 1 billion MC steps. Here, we use for each PE or PP block the optimum Δnpair value we estimated from the simulations of the corresponding PE and PP homopolymers, shown in Table 2. In Figure 9a−c, we plot with black squares the characteristic ratio of the block copolymer, CPE‑b‑PP , as a function of xPP for n pm = 0, 0.5, and 1.0, respectively. Here, xPP corresponds to the mole fraction of the PP block backbone bonds: nPP x PP = nPP + nPE (1)

Figure 7. Temperature coefficient, κ, versus the probability of meso dyads, pm, of a series of PP chains. Black squares correspond to our simulation results and red circles to the experimental measurements using SANS.95,110 Colored lines correspond to estimations from various RIS theoretical works and in particular the pink line to Mark,86 the brown line to Biskup et al.,73 the orange line to Suter and Flory,74 the blue line to Allegra et al.,108 and the green line to Alfonso et al.109

where nPP and nPE are the numbers of PP and PE block backbone bonds, respectively, and n = nPP + nPE is the number of backbone bonds in total. We have also added the estimated H


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Macromolecules

Figure 9. Characteristic ratio, C∞, as a function of the mole fraction of the PP block backbone bonds, xPP, for PE-b-PP copolymer chains with (a) syndiotactic (pm = 0), (b) atactic (pm = 0.5), and (c) isotactic (pm = 1.0) propylene dyads. Black squares correspond to our estimates of Cn from the MC simulations we performed. Red circles correspond to the estimated values using eq 5. Blue solid lines correspond to the predictions of the theoretical model we developed in eq 6. Green triangles in the insets of (a−c) correspond to the quantity ⟨RPE·RPP⟩/(nl2) versus xPP. (d) Dependence of ⟨RPE·RPP⟩/(nl2) on the molar fraction of the isotactic PP block backbone bonds, xPP, for various copolymer chain lengths.

C∞ values of PE (xPP = 0) and of s-PP, a-PP, or i-PP (xPP = 1) which correspond to pm = 0, 0.5, and 1.0, respectively. We should note that for xPP = 0 where nPE = n the characteristic PE ratio of the PE block CPE nPE = C∞ . In addition, for xPP = 1, where PP nPP = n, the characteristic ratio of the PP block CPP nPP = C∞ for all three tacticities. However, as we will discuss below, for 0 < PP xPP < 1 the characteristic ratios CPE nPE and CnPP of the PE and PP PE PE PP PP blocks satisfy CnPE ≤ C∞ and CnPP ≤ C∞ , respectively, because PP PE and PP blocks are not long enough and both CPE nPEand CnPP have not reached a plateau value yet. In all Figure 9a−c, we notice a monotonic increase or decrease of CPE‑b‑PP as a function of xPP, depending on whether n PP CPE ∞ is smaller or greater than C∞ . For example, in Figure 9a, PE s‑PP PE‑b‑PP C∞ 0.2 they begin to separate from each other. This effect indicates that for xP > 0.2 the tacticity of the propylene unit sequences appearing along the EPM chains begins to affect stiffness and CEPM starts ∞ PP value. To further approaching the corresponding C∞ investigate this effect, we plot in Figure 10b the fractions of PP dyads, triads, tetrads, and pentads along the EPM chains, f 2, f 3, f4, and f5, respectively, as a function of xP. We define fq as the fraction of the number of propylene q-ads (dyad, triad, tetrad, pentad, etc.) along an EPM chain divided by the maximum number of propylene q-ads which could appear along the EPM chain for a given xP. The latter maximum number equals the number of propylene q-ads present in a block copolymer of the same xP where all PP units are adjacent. We note that all fractions increase monotonically as xP increases. However, for xP ≤ 0.2, f 3, f4, and f5, denoted by red circles, green triangles, and blue diamonds, respectively, are very close to zero and only begin to increase for xP > 0.2, indicating that tacticity begins to affect EPM chain stiffness as soon as PP sequences which consist of three, four, or five propylene units emerge. Furthermore, the fq fractions correspond to the probability of having q − 1 sequential propylene units immediately after a given propylene unit along an EPM chain. With Bernoullian statistics, one would expect fq = xPq−1. In Figure 10b we plot with black, red, green, and blue solid lines the functions fq = xPq−1 for q = 2, 3, 4, and 5, respectively; we note that they are in perfect agreement with our simulation results.

nl

plotted with green triangles against xPP and we observe that ⟨R PE·R PP⟩ > 0. nl 2 To examine if the latter is a system size effect, we have also performed additional MC simulations on systems with n = nPE + nPP = 500, 1500, and 2000 backbone bonds. In Figure 9d, displaying the results of these simulations in the case of isotactic PP blocks, we note that, as we increase n, ⟨RPE·RPP⟩ → 0, which means that the end-to-end vectors for very long PE and PP blocks become uncorrelated. Under these conditions, we can conclude that PE‐b‐PP PE PP C∞ = (1 − x PP)C∞ + x PPC∞

(7)

In Figure 9a−c, we plot with blue lines eq 7, which finally corresponds to our predictions of CPE‑b‑PP for the PE-b-PP ∞ systems. EPMs and alt-PEP. The last systems we study are five series of EPM chains with pm = 0, 0.3, 0.5, 0.7, and 1 as well as an alternating ethylene−propylene copolymer (alt-PEP) chain. Each EPM series consists of 9 EPM chains with xP = 0.1, 0.2, ..., 0.9 per mole of propylene units. All EPM and alt-PEP chains are chosen to have the same backbone length with the PE-b-PP ones, i.e., n = 1000. In the beginning, a series of EPM chains with xP = 0.1, 0.2, ..., 0.9 with pm = 1 are generated, using the amorphous builder of MAPS suite.89 Then, using a home-built program, we generated the remaining four EPM series with pm = 0, 0.3, 0.5 and 0.7, by simply rotating the methyl branches until the desired tacticity was achieved. Therefore, the sequence of ethylene and propylene units for a given xP is the same for all pm. Single-chain MC simulations are performed on all EPM and alt-PEP chains for approximately 1 billion steps using the MC algorithm of Tzounis et al.93 for various Δnpair values to determine the optimum ones which maximize the stiffness of the chains. In Table 6, we present the optimum values of Δnpair for each pm, and xp for EPM chains. Table 6. Δnpair Values That Maximize CEPM ∞ for Each pm and xp for EPM Chains pm xP

0.0

0.3

0.5

0.7

1.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 3 3 3 J


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Macromolecules

Figure 10. (a) Characteristic ratio, CEPM ∞ , for different EPM chains plotted as a function of the propylene content xP. Red lines and symbols correspond to the system where all PP dyads are syndiotactic (pm = 0), black to the system where all PP dyads are isotactic (pm = 1), and purple, blue, and green lines and symbols correspond to the systems with pm = 0.3, 0.5, and 0.7, respectively. The orange star corresponds to the stiffness of an alt-PEP chain (pm = 0.5). (b) Fraction of PP dyads, triads, tetrads, and pentads, f 2, f 3, f4, and f5, respectively, along an EPM chain plotted as a function of propylene content, xP.

for both temperatures ⟨R2⟩0/M decays exponentially with wp, i.e., ⟨R2⟩0/M = A exp(−BwP) where A and B are free parameters. From the slopes of the fitted lines in Figure 11 we can estimate the pre-exponential factors A and the exponents B. As we note in Table 6, the exponents B estimated from our simulations are in very good agreement with the corresponding experimental results at both temperatures. However, all ⟨R2⟩0/ M values from our simulations are systematically higher than the experimental ones. The latter is also reflected in the values of A extracted from our simulations, which are higher than the experimental ones. As we also discussed in the previous sections, this effect is attributable to TraPPE force field90−93,101−103 we apply and more specifically to its torsional potential which leads to stiffer chain conformations than in reality.92,93,101−103

Table 7. Pre-Exponential Factor (A) and Exponent (B) Estimated by Fitting Equation ⟨R2⟩0/M = A exp(−BwP) to Both Simulation and Experimental Results of ⟨R2⟩0/M in Figure 11 as a Function of wP method

T (K)

MC simulations SANS experiments23 MC simulations SANS experiments58

348 348 413 413

B (Å2 mol g−1)

A 1.74 1.33 1.58 1.25

± ± ± ±

0.02 0.01 0.01 0.01

0.74 0.68 0.72 0.65

± ± ± ±

0.02 0.01 0.01 0.02

Finally, in Figure 11 we plot in a semilog the unperturbed chain dimensions, expressed as a fraction of the mean-square

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CONCLUSIONS In this work we have performed a systematic study of the effects of tacticity on the unperturbed dimensions of polypropylene homopolymer systems as well as of ethylene− propylene copolymers. The first systems we studied were PP homopolymers with Bernoullian tacticity sequences and various feactions of meso dyads, pm. We found that the stiffness of PP chains expressed by Flory’s characteristic ratio, C∞, displays a nonmonotonic behavior, having a minimum value at pm = 0.7. Our results are in qualitative agreement with previous estimates from RIS calculations. In addition, by analyzing the torsional states of skeletal bonds, we identified the two competing processes which act simultaneously and cause the nonmonotonic dependence of C∞ versus pm. Increasing pm, on one hand, decreases the length of all-trans sequences along the chain, leading to a reduction in stiffness; on the other hand, it increases the length of helical tgtg... sequences, which contributes to an increase in stiffness. Next, we studied how tacticity and propylene content affect the stiffness of PE-b-PP copolymer chains, extracting a general relation that connects the stiffness of the PE-b-PP chains with the propylene content. The characteristic ratio of the block copolymer was found to be linear in the mole fraction of propylene units, xP, in the limit of very long chains, for all PP block tacticities examined. Finally, we also studied how both tacticity and propylene content affect the stiffness of EPM

Figure 11. Semilog plot of the fraction of the mean-square end-to-end distance of a polymer chain, ⟨R2⟩0, divided by the molar mass of the chain M, for the PE, alt-PEP, and a-PP chains as a function of the propylene weight fraction, wP. Black lines and symbols correspond to the simulation results at both T = 348 and 413 K. Red lines and symbols correspond to SANS experimental measurements by Fetters et al.23,58 at both T = 348 and 413 K. Solid and dashed lines correspond to exponential decay fits to the corresponding simulation or experimental data.

end-to-end distance of a polymer chain, ⟨R2⟩0, divided by the molar mass of the chain, M, for PE, alt-PEP, and a-PP chains, versus the propylene weight fraction wp, at two different temperatures, T = 348 and 413 K. We compare our simulation results against the available experimental results. We note that K


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Macromolecules

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chains. We compared our results with the theoretical predictions of RIS models, and we found that they are in very good qualitative agreement; we also detected how the emergence of PP sequences longer than dyads along the EPM chains makes their stiffness sensitive to tacticity. The simulation and analysis methodology adopted in this study could be readily extended to future studies of the effects of tacticity and composition on the conformational properties of more complicated chiral polymers and copolymers such as polystyrene (PS) or poly(methyl methacrylate) (PMMA). In addition, one could examine the tacticity effects on physical properties such as the glass transition temperature or the selfdiffusion coefficient using for example molecular dynamics simulations.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (D.N.T.). ORCID

Panagiotis-Nikolaos Tzounis: 0000-0001-7824-6463 Stefanos D. Anogiannakis: 0000-0001-9766-2243 Doros N. Theodorou: 0000-0002-4763-9739 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Professor Apostolos Kyritsis for fruitful discussions. Scienomics SARL is gratefully acknowledged for making their software available to us through the Scienomics Group of Scientific Excellence (SGSE) program. We are grateful to Mr. Vasilios D. Georgilas for his help with the generation of the initial configurations of EPM systems.

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REFERENCES

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